1. Three conceptions of emergence: Material (ontological) Objective unpredictability Conceptual

2. One kind of material emergence – fusion Key idea: original objects lose their identity in fusing to produce a new object with novel properties Examples: Non-separable (entangled) states in quantum mechanics Some functional systems such as monetary units (but here properties are additive); a few merged organizations – e.g. UVA environmental science department

3. Unpredictability as computational incompressibility: One difference is between: (1) {S i (t+T)} = U{S i (t)}, for arbitrary T, where U is an updating function e.g. predicting a solar eclipse at any point in the future and (2) {S i (t+T)} = U T {S i (t)} e.g. unavoidably processing step-by-step iterations of updating for many agent based models.

4. Computer simulations produce computationally emergent phenomena. `Assume that P is a nominally emergent property possessed by some locally reducible system S. Then P is weakly emergent if and only if P is derivable from all of S’s micro facts but only by simulation.’ (Mark Bedau `Downward Causation and Autonomy in Weak Emergence’, Principia Revista Inernacional de Epistemologica 6 (2003), pp. 5-50.)

5. Computing the partition functions (and hence the exact energy levels) for finite lattices of non-planar 3D and 2D Ising models are NP-complete problems. (S. Istrail, Statistical Mechanics, Three-Dimensionality and NP-Completeness: I. Universality of Intractability of the Partition Functions of the Ising Model Across Non-Planar Lattices, Proceedings of the 32nd ACM Symposium on the Theory of Computing (STOC00), ACM Press, p. 87-96, Portland, Oregon, May 21-23, 2000 )

6. An algorithm is intractable if the number of computational steps or the amount of memory required for an output is an exponential function of the input. Example of intractable problem: For a statement of Presburger arithmetic, length n, every algorithm deciding its truth has a runtime of at least 2^[2^[cn]] for some constant c.

7. Conceptual Emergence `The ability to reduce everything to simply fundamental laws does not imply the ability to start from those laws and reconstruct the universe….The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity…..At each stage entirely new laws, concepts, and generalizations are necessary’ (P.W. Anderson `More is Different’, Science 177 (1972), pp.393)

8. One aspect Let X be the state space for the system S. Then: In general we do not have a closed form function F : X -> X that takes the current global state X(t) = (s 1 (t),...,s n (t)) 0 X and maps it into some other state in the state space X F(X(t)) = X(t+1) Such a function is only implicitly given through the updating schedule. [This is because there is only an implicit global state.] The classical dynamical systems which can be explicitly given such a global dynamics are special cases. (Modified from: Steen Rasmussen and Chris Barrett `Elements of a Theory of Simulation’ European Conference on Artificial Life ‘95)

9. Claims: 1.This re-conceptualization is often at the level of abstract, computationally tractable, structure. 2.Computational methods can thus result in a shifting of the boundaries between the sciences. What is important is computational form and methods (`computational templates’) not content. Fitness/energy landscapes are used to solve optimization problems in evolutionary biology, neural nets, financial markets; models originally invented to model the percolation of water through rocks are used to simulate forest fires, etc.

10. Some resources: `Emergence’ in The Encyclopedia of Philosophy (Second Edition). London: MacMillan, 2006. Emergence: Contemporary Readings in Science and Philosophy, Mark Bedau and Paul Humphreys (eds). Cambridge, MA: The MIT Press, 2007.